The two questions are, find the rule connecting C and H, the other one is, Find the cost of a job taking

Let's break down the problem and address each question accordingly.

### a) **Find the rule connecting $ C $ (Cost) and $ H $ (Hours)**

To find the rule connecting the cost $ C $ and the number of hours $ H $, we need to determine if there's a mathematical relationship, such as a linear equation, between the hours worked and the cost.

Given data from the table:

- $ H = 1 $ hour, $ C = 70 $
- $ H = 2 $ hours, $ C = 115 $
- $ H = 3 $ hours, $ C = 150 $
- $ H = 4 $ hours, $ C = 180 $
- $ H = 5 $ hours, $ C = 200 $

Let's assume a linear relationship of the form:

\[
C = mH + b
\]

Where:
- $ m $ is the slope (rate per hour),
- $ b $ is the y-intercept (the fixed cost).

To find $ m $ (the rate per hour), we can use the difference in cost for each hour:

- From $ H = 1 $ to $ H = 2 $: $ m = \frac{115 - 70}{2 - 1} = 45 $
- From $ H = 2 $ to $ H = 3 $: $ m = \frac{150 - 115}{3 - 2} = 35 $
- From $ H = 3 $ to $ H = 4 $: $ m = \frac{180 - 150}{4 - 3} = 30 $
- From $ H = 4 $ to $ H = 5 $: $ m = \frac{200 - 180}{5 - 4} = 20 $

It appears the rate per hour $ m $ decreases as the number of hours increases, meaning the relationship isn't perfectly linear. However, for simplicity, we can find the average rate or use the data to estimate a linear model:

Let’s calculate the average rate $ m $ using the first and last points:

\[
m = \frac{C_5 - C_1}{H_5 - H_1} = \frac{200 - 70}{5 - 1} = \frac{130}{4} = 32.5 \text{ dollars/hour}
\]

To find $ b $, we can use the point when $ H = 1 $ (assuming linearity):

\[
70 = 32.5(1) + b \implies b = 70 - 32.5 = 37.5
\]

So, the approximate linear equation relating $ C $ and $ H $ is:

\[
C = 32.5H + 37.5
\]

### b) **Find the cost of a job taking 2.5 hours and 3.5 hours**

Using the linear equation derived:

- For $ H = 2.5 $ hours:
  \[
  C = 32.5(2.5) + 37.5 = 81.25 + 37.5 = 118.75 \text{ dollars}
  \]

- For $ H = 3.5 $ hours:
  \[
  C = 32.5(3.5) + 37.5 = 113.75 + 37.5 = 151.25 \text{ dollars}
  \]

### Summary:
- **Rule connecting $ C $ and $ H $**: $ C = 32.5H + 37.5 $
- **Cost for**:
  - 2.5 hours: \$118.75
  - 3.5 hours: \$151.25

Would you like more details or have any questions? 

Here are 5 relative questions:
1. What if the relationship between cost and hours isn't linear—how would that change the equation?
2. How can we refine the model if we assume the rate decreases as hours increase?
3. What could the rate per hour be if we used different pairs of points for $ C $ and $ H $?
4. If a job took 6 hours, what would the predicted cost be using this linear model?
5. How would you approach finding a quadratic or exponential relationship between $ C $ and $ H $?

**Tip:** When modeling costs, always consider whether the relationship is strictly linear or if there are diminishing returns, which might suggest a different type of equation (e.g., quadratic or piecewise linear).

Explore the linear relationship between cost and hours worked. Learn how to find the rule connecting cost (C) and hours (H), and calculate costs for specific durations. Suitable for high school students studying linear equations and rates of change.

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